Animate dog character with soft body physics "jello" legs?
If I want to combine soft body physics with an armature, I usually do it via a dynamic mesh deform:

Suzanne has an armature modifier followed by a dynamically-bound mesh deform modifier.  The target of the mesh deform modifier is bone parented to an appropriate bone-- here, to Suzanne's ear bone-- and has soft body physics, with a goal group and the default goal strength set to 1.0.  There is plenty you can do to further tweak the physics.  The quality of the mesh deform can be increased by using a vertex group to limit it, or by creating an outer shell with full goal, to smoothly interpolate from physics-based deformation to no deformation.
An alternative here is to track the positions of soft-body vertices (or rigid body origins) with your armature.  For example:

Here, I've marked vertices in the soft body with vertex groups that the armature uses for damped track and locked track constraints, so that the bone automatically follows what the physics does (to a certain extent).  The soft body in this case is not bone parented to the ear bone, but to the ear bone's parent.  This can be done with rigid body physics as easily as with soft body physics, by tracking the origins of rigid body objects instead of marked vertex groups.